12-2004
A multienvironment conditional probability
density function model for turbulent reacting flows
Rodney O. Fox
Iowa State University, [email protected]
Venkatramanan Raman
Stanford University
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reacting flows
R. O. Fox and V. RamanCitation: Physics of Fluids (1994-present) 16, 4551 (2004); doi: 10.1063/1.1807771
View online: http://dx.doi.org/10.1063/1.1807771
View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/16/12?ver=pdfcov Published by the AIP Publishing
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A multienvironment conditional probability density function model
for turbulent reacting flows
R. O. Foxa)
Department of Chemical Engineering, Iowa State University, Ames, Iowa 50011-2230 V. Ramanb)
Center for Turbulence Research, Stanford University, Stanford, California 94305
(Received 29 April 2004; accepted 26 August 2004; published online 5 November 2004)
The multienvironment conditional probability density function(MECPDF) model was first proposed by Fox [Computational Models for Turbulent Reacting Flows (Cambridge University Press, Cambridge, 2003)] as a simple extension of multienvironment probability density function models for turbulent reacting flows. Like the conditional moment closure (CMC) and the laminar flamelet model (LFM), the MECPDF model describes the reacting scalars conditioned on the value of the mixture fraction. However, unlike CMC and LFM, the new model provides a consistent description of conditional fluctuations in both the scalar dissipation rate and the reacting scalars, and hence can be used to model partial extinction and reignition in homogeneous turbulent reacting flows. In this work, a general derivation of the MECPDF model is presented for a single reaction-progress variable using the direct quadrature method of moments. Extensions of the model to multiple reaction-progress variables and conditioning on the mixture-fraction vector are also discussed. After deriving the model, the closure assumptions are validated using direct simulations for pure diffusion of two randomly distributed, initially correlated scalar fields. Two homogeneous applications are then considered: nonreactive mixing starting from nontrivial initial conditions, and reactive mixing with partial extinction and reignition. © 2004 American Institute of Physics.
[DOI: 10.1063/1.1807771]
I. INTRODUCTION
The turbulent mixing of chemically reacting scalars is a problem of great interest in many fields of science and technology.1 In the field of turbulent combustion, problems of particular interest are nonpremixed and partially premixed flows.2At moderate to high Reynolds numbers, such flows are known to exhibit partial extinction and reignition of local flame structures.3 Turbulent combustion models for fully burning flames, and flames that exhibit extinction are well understood and widely used in practical calculations.2On the other hand, models that can capture both extinction and re-ignition are more difficult to formulate because they should account for the complex interactions between local fluctua-tions in turbulent mixing (that lead to extinction) and local flame structures (that lead to reignition).4–6 Nevertheless, based on recent direct-numerical simulations (DNS),5–7 it now appears that two key elements are required for a suc-cessful model:(1) a representation of the probability density function (PDF) of the scalar dissipation rate, and (2) a mechanism for interactions between “flamelet” and “non-flamelet” structures in the flow.
Fluctuations in the scalar dissipation rate are known to be significant in turbulent flows,8 and large fluctuations can lead to local extinction in nonisothermal reacting flows.5,7
Once extinguished, local fluid elements can be reignited by diffusive mixing with neighboring flame structures.5 Many successful models for representing extinction in nonpre-mixed turbulent flames [e.g., the laminar flamelet model9
(LFM) and the conditional moment closure10
(CMC)] are
derived by conditioning on the mixture fraction.1,2Thus, for example, the flamelet model predicts extinction when the local value of the mixture-fraction dissipation rate is larger than a critical “quench” value. However, once quenched, an isolated flamelet has no mechanism for reignition in station-ary turbulence; hence the need to include a model for the interactions between fluid elements.4,6 Moreover, once quenched, the assumption of a quasisteady state between mo-lecular diffusion and chemical reactions used to drop the spatial transport terms in the flamelet model2 is no longer valid.6 In the context of the CMC model, a similar break-down occurs at local extinction where the conditional vari-ance of the reaction-progress variable is no longer negligible.5,7 In this so-called “distributed-combustion” re-gime, turbulent combustion models based on transported PDF methods are much more successful.1,11Likewise, mod-els such as the Lagrangian modified flamelet model6 that account for extinction due to the fluctuating scalar dissipa-tion rate and reignidissipa-tion due to “interacdissipa-tions” between burn-ing and nonburnburn-ing flamelets also show great promise.
The goal of the present work is to develop a conditional PDF model for inhomogeneous turbulent reacting flows that overcomes the shortcomings of existing models (i.e., both the micromixing closures used in transported PDF methods
a)Telephone: (515) 294-9104. Fax: (515) 294-2689. Electronic mail:
b)Telephone: (650) 725-6635. Fax: (650) 725-3525. Electronic mail:
1070-6631/2004/16(12)/4551/15/$22.00 4551 © 2004 American Institute of Physics
and conditional models). In order to focus on the ability of the model to describe interactions between “burning” and “nonburning” regions in composition space, we limit our-selves to consideration of a single reaction-progress variable conditioned on the mixture fraction. The derivation of the model begins with the joint PDF transport equation as de-scribed in Sec. II. Because we are interested in obtaining a consistent model for inhomogeneous reacting flows, the spa-tial transport terms in the PDF transport equation are closed using a standard gradient-diffusion model.1,11Other consis-tent models could also be used to close the velocity fluctua-tion term and would not affect the principal conclusions drawn in this work. On the other hand, the terms represent-ing mixrepresent-ing due to molecular diffusion are unclosed and must be modeled by invoking assumptions similar to those used in deriving the CMC model.12
The details of the derivation using the direct quadrature method of moments1,13(DQMOM) are discussed in Sec. III, where the final forms of the inhomogeneous transport equa-tions for the multienvironment conditional PDF(MECPDF) model are given. Quadrature-based moment methods are a powerful technique13–16 for approximating with controllable accuracy the moments of a distribution function(e.g., trans-ported PDF or number density functions) starting from its transport equation. In recent work on isothermal reacting flows,16 DQMOM has been shown to yield results for the lower-order scalar moments that are in excellent agreement with transported PDF methods at a fraction of the computa-tional cost. In the present work, we extend DQMOM to treat the conditional PDF of a reaction-progress variable in order to describe nonisothermal reacting flows. In Sec. IV, we fo-cus on a two-environment homogeneous model that de-scribes the conditional mean and variance of the reaction-progress variable. In Sec. V, we use data for pure diffusion of randomly distributed scalar fields to validate the closures in-troduced in Sec. III. We then apply the homogeneous two-environment conditional PDF model to study nonreactive scalar mixing with nontrivial initial conditions, and reactive scalar mixing with partial extinction and reignition. Conclu-sions are drawn in Sec. VI.
II. CONDITIONAL PDF MODELS
The homogeneous MECPDF model was proposed by Fox1as an ad hoc extension of the conservative form of the conditional moment closure.12 The MECPDF model is de-rived starting from the following transport equation for the joint PDF of a reaction-progress variable Y and the mixture-fractionwherein the spatial transport terms due to the fluc-tuating velocity have been closed using a gradient-diffusion model: fY, t +具Ui典 fY, xi = xi
冉
⌫t fY, xi冊
− y关SY共y,兲fY,兴 −1 2 2具⑀ Y兩y,典fY, y2 − 2具⑀ Y兩y,典fY, y −1 2 2具⑀ 兩y,典fY, 2 . 共1兲In this expression,具Ui典 is the mean velocity, ⌫tis the turbu-lent diffusivity, SY is the chemical source term for Y, and
具⑀ij兩y,典 are the (unknown) joint scalar dissipation rates
con-ditioned on Y = y and=. The joint scalar dissipation rates are defined by1 ⑀ij= 2⌫ i xk j xk, 共2兲
where⌫ is the molecular diffusivity (assumed to be equal for both scalars 1= Y and 2=). Integrating Eq. (1) over
reaction-progress space yields the transport equation for the mixture-fraction PDF, f t +具Ui典 f xi = xi
冉
⌫t f xi冊
−1 2 2具⑀ 兩典f 2 , 共3兲where 具⑀兩典 is the mixture-fraction scalar dissipation rate conditioned on=.
Using well-established techniques,12 Eq.(1) can be ma-nipulated to find the CMC transport equation for the condi-tional reaction-progress variable具Y 兩典 in conservative form,
具Y兩典f t +具Ui典 具Y兩典f xi = xi
冉
⌫t 具Y兩典f xi冊
+ SY共具Y兩典,兲f +1 2 冉
具⑀兩典f 具Y兩典 −具Y兩典 具⑀兩典f 冊
, 共4兲or[using Eq. (3)] in nonconservative form,
具Y兩典 t +具Ui典 具Y兩典 xi = xi
冉
⌫t 具Y兩典 xi冊
+2⌫t f f xi 具Y兩典 xi + SY共具Y兩典,兲 + 1 2具⑀兩典 2具Y兩典 2 . 共5兲As discussed elsewhere,1具⑀兩典 and fmust be chosen such that they satisfy Eq.(3), in which case Eq. (5) will conserve the scalar mean具Y典 when SYis null. In contrast, Eq.(4) will
conserve the scalar mean in the nonreactive limit for any choice of 具⑀兩典 and f that satisfies appropriate boundary conditions at= 0 and 1. We note, however, that conserva-tion of the scalar mean does not automatically imply that
具Y 兩典 will be physically realizable (i.e., remain within the
convex hull defined by the initial conditions). In fact, our experience with using the homogeneous form of Eq.(4) for the nonreactive case with the initial conditions described in Sec. IV has shown that realizability can only be attained if
具⑀兩典 and fsatisfy Eq.(3). Thus, when properly employed
to ensure conservation and realizability, Eqs.(4) and (5) will yield identical results.
The key assumptions that are used to derive Eq.(4) are the following. First, the conditional PDF of Y given =共i.e., fY兩= fY,/ f兲 is assumed to be a␦ function:
fY兩共y兩兲 =␦共y − 具Y兩典兲. 共6兲
It then follows that 具SY兩典=SY共具Y 兩典,兲. Next, the
condi-tional joint scalar dissipation rate is assumed to obey
具⑀Y兩典 =
冕
具⑀Y兩y,典fY兩dy =具⑀兩典 具Y兩典 , 共7兲
which is consistent with using Y共x,t兲=具Y 兩共x,t兲典 in Eq. (2). As discussed elsewhere,1 the homogeneous version of Eq.(5) has the same form as the unsteady laminar flamelet model.2The principal difference is that in the laminar flame-let model具⑀兩典 is multiplied by a random variable in order to represent fluctuations in the conditional scalar dissipation rate. Thus, the laminar flamelet model can describe extinc-tion due to fluctuaextinc-tions in the mixing intensity. However, the laminar flamelet model ignores interactions between flame-lets and thus cannot describe the reignition of a flamelet due to diffusive mixing. In contrast, the CMC model represents mixing by a single characteristic time scale associated with
具⑀兩典, and thus cannot describe local extinction and
reigni-tion events. With these shortcomings in mind, one of the primary motivations for introducing the MECPDF model will be to describe local extinction in turbulent reacting flows. Recently, the Lagrangian modified flamelet model6 was developed to account successfully for local flame extinc-tion and reigniextinc-tion. The MECPDF shares some similarities with this model, but differs in other important aspects as discussed in Sec. III C.
In the context of multienvironment models, Eq.(6) rep-resents a one-environment model for fY兩. In the MECPDF
model, we generalize the assumed form of the conditional PDF to multiple environments,
fY兩共y兩兲 =
兺
n=1 N
pn共兲␦共y − 具Y兩典n兲, 共8兲 where pn共; x , t兲 is the probability of environment n, and
具Y 兩典n共x,t兲 is the conditional reaction-progress variable in
environment n. Note that 具Y 兩典n cannot be found directly
from the conditional PDF, but rather is defined by forcing a selected set of conditional moments to agree with their defi-nition from the conditional PDF. Thus, Eq.(8) can be under-stood as a quadrature approximation14 of order N for the conditional PDF that is consistent with a given set of condi-tional moments. Indeed, given Eq.(8), higher-order tional moments can be computed. For example, the condi-tional first and second moments are
具Y兩典 =
兺
n=1 N pn共兲具Y兩典n 共9a兲 and 具Y2兩典 =兺
n=1 N pn共兲具Y兩典n 2. 共9b兲Thus, if N = 2 there are four unknowns: p1, p2, 具Y 兩典1, and
具Y 兩典2, which can in principle be determined from an equal
number of conditional moments:具Yk兩典 with k=0, 1, 2, 3. As
with other quadrature methods,14,15 the accuracy of the ap-proximation increases rapidly with increasing N. Based on the quadrature approximation, the conditional chemical source term can be expressed as
具SY兩典 =
兺
n=1 Npn共兲SY共具Y兩典n,兲. 共10兲 Thus, for 2艋N, the MECPDF model provides a description of the conditional variance, which will be useful for describ-ing fluctuations about the conditional mean. Moreover, be-cause the conditional scalar dissipation rate for each environ-ment 具⑀兩典n can be different, the MECPDF model will be
able to describe local extinction in environment n. In the following section, we derive transport equations for wn
= pnfand wn具Y 兩典nstarting from Eq. (1) using DQMOM.
III. DERIVATION OF THE MECPDF MODEL
The original ad hoc derivation1 of the MECPDF model did not make use of the transport equation for the joint PDF of Y and . A rigorous derivation using DQMOM (Ref. 1) starts with the joint PDF written in the form
fY, t +具Ui典 fY, xi − xi
冉
⌫t fY, xi冊
= P共y,兲, 共11兲 where P共y,兲 = − y关SY共y,兲fY,兴 − 1 2 2具⑀ Y兩y,典fY, y2 − 2具⑀ Y兩y,典fY, y − 1 2 2具⑀ 兩y,典fY, 2 . 共12兲As shown below, the terms on the left-hand side of Eq.(11) yield transport equations of the form
wn t +具Ui典 wn xi − xi
冉
⌫t wn xi冊
= an 共13兲 and wn具Y兩典n t +具Ui典 wn具Y兩典n xi − xi冉
⌫t wn具Y兩典n xi冊
= bn, 共14兲where an共兲 and bn共兲 are source terms that are found from the conditional moments of P共y,兲. Before applying DQ-MOM, we should note that, unlike in earlier applications where closed-form PDF transport equations were used,1,16 the joint scalar dissipation rates in Eq.(12) do not appear in closed form. It will thus be necessary to introduce consistent modeling assumptions to close these terms. In Sec. V, we will explore the validity of these assumptions using direct simulations of the scalar diffusion equation for two corre-lated scalar fields that are initialized as random lamellar sys-tems(RLS).17
A. Space and time derivatives
In the MECPDF model, the joint PDF is represented by
fY,共y,兲 =
兺
n=1 Nwn共兲␦共y − 具Y兩典n兲, 共15兲
and the weights wnand abscissas具Y 兩典nare found by forcing
them to be consistent with the moments of the joint PDF. In
the DQMOM approach, this expression is substituted into the left-hand side of Eq.(11) to find the transport terms in Eqs.
(13) and (14). Starting with the convective terms, we find
DfY, Dt = fY, t +具Ui典 fY, xi =
兺
n=1 N冋
Dwn Dt册
␦共y − 具Y兩典n兲 −兺
n=1 N冋
wn D具Y兩典n Dt册
␦ 共1兲共y − 具Y兩典 n兲, 共16兲 or DfY, Dt =n=1兺
N冋
Dwn Dt册
␦共y − 具Y兩典n兲 −兺
n=1 N冋
Dwn具Y兩典n Dt −具Y兩典nDwn Dt册
␦ 共1兲共y − 具Y兩典 n兲, 共17兲where␦共k兲denotes the kth derivative of the␦ function.18 The spatial diffusion term in Eq.(11) can be computed in two steps. First, the spatial derivative is written as
fY, xi =
兺
n=1 N冋
wn xi册
␦共y − 具Y兩典n兲 −兺
n=1 N冋
wn具Y兩典n xi −具Y兩典n wn xi册
␦共1兲共y − 具Y兩典 n兲. 共18兲The overall term then becomes
xi
冉
⌫t fY, xi冊
=兺
n=1 N冋
xi冉
⌫t wn xi冊
册
␦共y − 具Y兩典n兲 −兺
n=1 N冋
xi冉
⌫t wn具Y兩典n xi冊
−具Y兩典n xi冉
⌫t wn xi冊
册
␦共1兲共y − 具Y兩典 n兲 +兺
n=1 N wncn␦共2兲共y − 具Y兩典n兲, 共19兲 where cn共兲 = ⌫t冉
具Y兩典n xi冊
2 . 共20兲Note that when solving the transport equations [Eqs. (13) and(14)] cn will be known. Also note that the form of the
term involving cnin Eq.(19) results from using the
gradient-diffusion model for the velocity fluctuations. If other models were to be used, the exact form of this term would differ.
Collecting together all of the terms in Eqs.(17) and (19) and using Eqs.(13) and (14), we find from Eq. (11) that
兺
n=1 N
关␦共y − 具Y兩典n兲 + 具Y兩典n␦共1兲共y − 具Y兩典n兲兴an−
兺
n=1 N ␦共1兲共y −具Y兩典n兲bn=兺
n=1 N ␦共2兲共y − 具Y兩典 n兲wncn+ P共y,兲, 共21兲where anand bnare the unknown source terms. By
comput-ing its conditional moments, this expression can be used to
generate a system of 2N linear equations for the source terms1 共1 − m兲
兺
n=1 N 具Y兩典n m an+ m兺
n=1 N 具Y兩典n m−1 bn = m共m − 1兲兺
n=1 N 具Y兩典n m−2w ncn+ Pm共兲, m = 0, . . . ,2N − 1, 共22兲 where the conditional moments in phase space are defined byPm共兲 =
冕
ymP共y,兲dy. 共23兲The next step is to find Pmstarting from Eq. (12).
B. Conditional moments in phase space
The four transport terms in phase space appearing in Eq.
(12) can be treated separately. Beginning with the drift term
in reaction-progress-variable space, we find
冕
ym y关SY共y,兲fY,兴dy = − m冕
y m−1S Y共y,兲fY,dy = − m冕
ym−1SY共y,兲 ⫻兺
n=1 Nwn␦共y − 具Y兩典n兲dy
= − m
兺
n=1 N wn具Y兩典n m−1 SY共具Y兩典n,兲. 共24兲For clarity, we have shown all of the steps in the manipula-tions. In the first line, we use integration by parts. In the second line, we substitute the assumed form of the joint PDF. Finally, in the last line we integrate using the properties of the␦ function.
The (unclosed) term involving 具⑀Y兩y,典 will be treated
next:
冕
ym 2具⑀ Y兩y,典fY, y2 dy = m共m − 1兲冕
ym−2具⑀Y兩y,典fY,dy = m共m − 1兲冕
ym−2具⑀Y兩y,典兺
n=1 Nwn␦共y − 具Y兩典n兲dy
= m共m − 1兲
兺
n=1 N
wn具Y兩典nm−2具⑀Y兩具Y兩典n,典. 共25兲 The exact form of具⑀Y兩具Y 兩典n,典 is unknown. Thus,
consis-tent with the flamelet and CMC models, we will assume that
具⑀Y兩具Y兩典n,典 = 具⑀Y兩典n=具⑀兩典n
冉
具Y兩典n
冊
2
, 共26兲 where each environment is assumed to have its own condi-tional scalar dissipation rate 具⑀兩典n. Note that the latter is required to describe local extinction of one environment due to high local mixing. However, as we shall discover later, the assumed form of Eq. (26) neglects diffusive mixing in Y space in the direction normal to 具Y 兩典n. [For example, if
具Y 兩典nis independent of, then Eq.(26) predicts zero
diffu-sive mixing.] Thus, it neglects micromixing between differ-ent environmdiffer-ents and cannot describe reignition. As done in the original model,1 it will be necessary in Sec. III C to add additional terms to account for micromixing in Y space. We will explore the validity of this and related assumptions in Sec. V.
The term involving具⑀Y兩y,典 will be treated next:
冕
ym 2具⑀ Y兩y,典fY, y dy = − m 冕
ym−1具⑀Y兩y,典fY,dy = − m 冕
y m−1具⑀ Y兩y,典兺
n=1 Nwn␦共y − 具Y兩典n兲dy
= − m
兺
n=1 N 共wn具Y兩典n m−1具⑀ Y兩具Y兩典n,典兲. 共27兲The exact form of具⑀Y兩具Y 兩典n,典 is unknown. Again,
consis-tent with the flamelet and CMC models, we will assume that
具⑀Y兩具Y兩典n,典 = 具⑀Y兩典n=具⑀兩典n 具Y兩典n
. 共28兲
The term involving具⑀兩y,典 will be treated next:
冕
ym 2具⑀ 兩y,典fY, 2 dy = 2 2冕
y m具⑀ 兩y,典fY,dy = 2 2冕
y m具⑀ 兩y,典兺
n=1 N wn⫻␦共y − 具Y兩典n兲dy
=
兺
n=1 N 2 2共wn具Y兩典n m具⑀ 兩具Y兩典n,典兲. 共29兲The exact form of具⑀兩具Y 兩典n,典 is again unknown.
Consis-tent with the flamelet and CMC models, we will assume that
具⑀兩具Y兩典n,典 = 具⑀兩典n. 共30兲
As with the CMC and flamelet models, the functional form of 具⑀兩典n must be specified by the user and be consistent with the presumed form of the mixture-fraction PDF. We will return to this issue in Sec. IV.
Collecting together all of the terms, Pmcan now be
writ-ten as Pm= m
兺
n=1 N wn具Y兩典nm−1SY共具Y兩典n,兲 − m共m − 1兲 2兺
n=1 N wn具Y兩典n m−2具⑀ 兩典n冉
具Y兩典n 冊
2 + 1 2兺
n=1 N 冉
wn具⑀兩典n 具Y兩典n m −具Y兩典n mwn具⑀兩典n 冊
. 共31兲By rewriting the final two terms, this expression can be writ-ten in a simpler form
Pm= m
兺
n=1 N wn具Y兩典n m−1冉
SY共具Y兩典n,兲 + 1 2具⑀兩典n 2具Y兩典 n 2冊
− 1 2兺
n=1 N 具Y兩典n m 2w n具⑀兩典n 2 . 共32兲This final expression for Pmcan now be used in Eq.(22) to
find the source terms.
C. Consistent source terms
The linear equation for the source terms [Eq. (22)] can be simplified by introducing two new unknown source terms an*and bn* defined in terms of anand bnby
an*= an+ 1 2 2w n具⑀兩典n 2 共33兲 and bn*= bn− wnSY共具Y兩典n,兲 −1 2
冉
wn具⑀兩典n 具Y兩典n −具Y兩典nwn具⑀兩典n 冊
. 共34兲Using these definitions, Eq.(22) becomes
共1 − m兲
兺
n=1 N 具Y兩典n ma n *+ m兺
n=1 N 具Y兩典n m−1b n * = m共m − 1兲兺
n=1 N 具Y兩典n m−2 wncn. 共35兲Thus, for homogeneous flow, we have cn= 0 so that an
* = bn* = 0, and wn t = − 1 2 2w n具⑀兩典n 2 共36兲 and 具Y兩典n t = SY共具Y兩典n,兲 + 1 2具⑀兩典n 2具Y兩典 n 2 , 共37兲
which have the same forms as Eqs.(3) and (5), respectively. Hence, as pointed out earlier, Eq.(37) predicts that the con-ditional reaction-progress variable follows the homogeneous CMC model without any mixing between environments.
In addition to neglecting micromixing between environ-ments, the source terms given above [i.e., Eq. (36)] imply that pn must be a function of. However, for the
homoge-neous case, we expect pnto be independent of. We can also
note that the -space transport term in Eq. (37) does not reproduce the CMC expression when summed over all n. This implies that the conditional mean具Y 兩典 will be incon-sistent with the CMC model. Note that this inconsistency will also result in the unconditional mean具Y典 not being con-served in the absence of chemical reactions. Thus, we can conclude that the present form of the source terms anand bn are inconsistent with the expected behavior.
Similar inconsistency problems arise in the Lagrangian flamelet model.6In order to conserve the mean, Mitarai, Ko-sály, and Riley6add an ad hoc linear term on the right-hand side of their Lagrangian modified flamelet model, and fix the coefficient by forcing the mean to be conserved. Likewise, they implement mixing between flamelets(equivalent to “en-vironments” in the present context) by enforcing ad hoc boundary conditions on extinguished flamelets. While these modifications solve the above-mentioned consistency prob-lems, they require preexisting knowledge of the extinction limit and thus are difficult to extend to a more general
frame-work(e.g., to treat nonreactive scalar mixing).
Here, in order to make the source terms consistent, we will leave the boundary conditions unchanged and simply add correction terms to Eqs.(36) and (37):
wn t = − 1 2 2w n具⑀兩典n 2 + pnGn 共38兲 and 具Y兩典n t = SY共具Y兩典n,兲 + 1 2具⑀兩典n 2具Y兩典 n 2 + Mn. 共39兲
These terms must be defined such that they do not change the mixture-fraction PDF[sum of Eq. (38) over all n],
兺
n=1 N
pnGn= 0, 共40兲
and such that the conditional mean is conserved during mix-ing inspace:
FIG. 1. Conditional statistics from direct simulation of scalar diffusion. Top left:具Y 兩典. Top right: 具⑀兩典/具⑀典. Bottom left: f. Bottom right: fY. Lines
correspond to particular values of the mixture-fraction standard deviation. Circle: 0.9. Square: 0.8. Diamond: 0.7. Up-triangle: 0.6. Left-triangle: 0.5. Down-triangle: 0.4. Right-triangle: 0.3.
兺
n=1 N pnMn= 1 2具⑀兩典 2具Y兩典 2 −兺
l=1 N 1 2pl具⑀兩典l 2具Y兩典 l 2 . 共41兲As in the Lagrangian modified flamelet model,6 the exact forms of the correction terms are unknown. However, if we assume that具⑀兩典n= hn具⑀兩典 where hn is independent of,
then pn being constant requires that
Gn=共hn− 1兲 1 2 2具⑀ 兩典f 2 , 共42兲
where, by definition of具⑀兩典 in terms of 具⑀兩典n,
兺
n=1 N
pnhn= 1. 共43兲
Note that, as with the CMC model, when applying the MECPDF model the functional forms of 具⑀兩典 and f are assumed to be known. Thus, the dependence of Gn on in
Eq.(42) will be known.
Determination of a suitable form for Mn is more
arbi-trary. Indeed, there are several different expressions that would satisfy the constraint in Eq.(41). For example, one of the simplest possible forms is
TABLE I. A priori statistics for a two-environment representation of pure-diffusion data.
具Y典1/具Y典 具Y典2/具Y典 h1 h2 1共0.5兲 2共0.5兲 CY
0.9 1.000 1.000 1.000 1.000 0.0 0.0 ¯ 0.8 0.807 1.193 1.429 0.571 0.18 0.25 2.02 0.7 0.709 1.291 1.521 0.479 0.26 0.34 1.42 0.6 0.667 1.333 1.467 0.533 0.31 0.39 1.03 0.5 0.683 1.317 1.364 0.636 0.30 0.38 0.98 0.4 0.733 1.267 1.307 0.693 0.25 0.32 1.03 0.3 0.797 1.203 1.199 0.801 0.20 0.26 1.07
FIG. 2. Conditional means from direct simulation of scalar diffusion. Top left:= 0.8. Top right:= 0.7. Bottom left:= 0.5. Bottom right:= 0.3. Lines correspond to particular models. Solid line:具Y 兩典. Empty circle: 具Y 兩典1
EA. Filled circle:具Y 兩典 2
EA. Empty square:具Y 兩典 1
QMOM. Filled square:具Y 兩典 2 QMOM.
Mn= 1 2具⑀兩典 2 2
冉
具Y兩典n−兺
l=1 N plhl具Y兩典l冊
+␥共具Y兩典 − 具Y兩典n兲. 共44兲The first term on the right-hand side ensures that the model reproduces the CMC model for具Y 兩典. The second term is a conditional version of the interaction-by-exchange-with-the-mean(IEM) model,1where␥controls the rate of micromix-ing between environments with different scalar dissipation rates. While other micromixing models1 developed for un-conditional scalars could equally well be employed, the simple form of the IEM model makes it attractive for a pre-liminary investigation of the MECPDF model and thus will be used here.
Note that the “diffusion” term in Eq.(44) can be either positive or negative. When employed in Eq.(39), the overall diffusion term for具Y 兩典n becomes
1 2具⑀兩典 2 2
冉
共1 + hn兲具Y兩典n−兺
l=1 N plhl具Y兩典l冊
. 共45兲 When multiplied by pn and summed over all environments,this term leads to the diffusion term in the CMC model for
具Y 兩典 as required for consistency. In order for the model to
be stable, the effective diffusion coefficient in Eq.(45) must be non-negative for all possible values of hn. We will show
that this is the case for the two-environment model in Sec. V. In conclusion, in order to make the source terms consis-tent in the homogeneous limit, we need additional terms on the right-hand sides of Eqs.(33) and (34). The final forms for the consistent source terms are
an= an * −1 2 2w n具⑀兩典n 2 + pnGn 共46兲 and bn= bn*+ pnSY共具Y兩典n,兲f+1 2pnhn
冉
具⑀兩典f 具Y兩典n −具Y兩典n具⑀兩典f 冊
+具Y兩典npnGn+ pnMnf, 共47兲where an*and bn*are found from Eq.(35), and Gnand Mnare given by Eqs.(42) and (44), respectively. These source terms are then used in Eqs.(13) and (14) to solve for pnand具Y 兩典n,
respectively.
FIG. 3. Conditional mixture-fraction dissipation rate from direct simulation of scalar diffusion. Top left:= 0.8. Top right:= 0.7. Bottom left:= 0.5. Bottom right:= 0.3. Solid line:具⑀兩典/具⑀典. Empty circle: 具⑀兩典1/具⑀典. Filled circle: 具⑀兩典2/具⑀典.
D. MECPDF transport equations
In summary, the inhomogeneous MECPDF model is de-fined by transport equations of the form
wn t +具Ui典 wn xi = xi
冉
⌫t wn xi冊
+ an*−1 2 2w n具⑀兩典n 2 + pnGn 共48兲 and wn具Y兩典n t +具Ui典 wn具Y兩典n xi = xi冉
⌫t wn具Y兩典n xi冊
+ bn*+ wnSY共具Y兩典n,兲 +1 2 冉
wn具⑀兩典n 具Y兩典n −具Y兩典n wn具⑀兩典n 冊
+具Y兩典npnGn+ wnMn, 共49兲 where an* and bn* are found from Eq. (35). Due to the negative-diffusion term inspace, Eq.(48) cannot be solved directly. Instead, it can be used to find a transport equation for pn. In order to facilitate this procedure, we will take an*
= 0. Note that this choice simplifies the equation for the source terms[Eq. (35)] at the expense of reducing the num-ber of conditional moments that can be accurately controlled by DQMOM from 2N to N. Thus, for example, with N = 2
only the conditional mean and conditional variance can be accurately predicted for inhomogeneous systems. Using this assumption yields pn t +共具Ui典 + Vi兲 pn xi = xi
冉
⌫t pn xi冊
, 共50兲 where Vi= − 2⌫tln共f兲 xi . 共51兲The N unknown terms bn*in Eq.(49) are then found from
兺
n=1 N 具Y兩典n m−1 bn*=共m − 1兲兺
n=1 N 具Y兩典n m−2 wncn 共52兲with m = 1 ,¯ ,N. Note that for homogeneous flows, cn= 0
and thus bn*= 0. Finally, we note that Eq. (49) appears in conservative form. Using Eq. (48), the corresponding non-conservative equation for 具Y 兩典n can be derived. As
dis-cussed in Sec. II for the CMC model, the results will be identical only if consistent forms are used for fand具⑀兩典. The DQMOM derivation of the MECPDF model dis-cussed above can be extended in two directions, which are as follows.
FIG. 4. Conditional joint dissipation具⑀Y兩典 from direct simulation of scalar diffusion. Top left:= 0.8. Top right:= 0.7. Bottom left:= 0.5. Bottom right:
= 0.3. Solid line:具⑀Y兩典/具⑀典. Empty circle: 具⑀Y兩典1/具⑀典. Filled circle: 具⑀Y兩典2/具⑀典. Dashed lines: model 具⑀兩典n共具Y 兩典n/兲/具⑀典.
(i) DQMOM can be used with multiple reacting
scalars.16Thus, the MECPDF model can be extended to mul-tiple reaction-progress variables conditioned on mixture fraction:1 具rp兩典.
(ii) DQMOM can be used with conditioning on the
mixture-fraction vector. For a single reaction-progress vari-able, this extension should be straightforward. The principal complication in practical applications is the fact that a closed form solution for f共兲 is not available.1 Thus, the corre-sponding PDF equation must be solved to find f. Analytical solutions for f would open the door to efficient computa-tional methods for describing mixing between multiple inlet streams with different compositions.
In the remainder of this work, we will consider only the homogeneous MECPDF model, which is given by Eqs.(38) and(39). This last expression has the form of an “interact-ing” flamelet model wherein具⑀兩典nis the scalar dissipation
rate for the nth flamelet. The choice of the micromixing models共Mn兲 will thus control the reignition properties of the
MECPDF model. Likewise, the choice of 具⑀兩典n= hn具⑀兩典
will control the extinction properties of environment n. In the following, we will consider only a two-environment condi-tional PDF model(i.e., N=2).
IV. TWO-ENVIRONMENT CONDITIONAL PDF MODEL
The multienvironment PDF model offers a low-cost al-ternative to solving Eq. (1) using Monte Carlo methods. In Wang and Fox,16it is shown that even with N = 2 the agree-ment with the Monte Carlo results for the lower-order mo-ments is very good. Thus, it is of interest to investigate a two-environment conditional PDF model as an extension of the CMC and flamelet models. Recall that with N = 2, the MECPDF model should provide an accurate quadrature ap-proximation for the conditional mean and conditional vari-ance[Eqs. (9a) and (9b)].
The first task is to specify hn共t兲, which should depend on
the shape of the PDF of the scalar dissipation rate. In par-ticular, for a two-environment model hn should depend on
TABLE II. MECPDF predictions with h1= 1.6 and CY= 1 for pure-diffusion
statistics in Table I.
具Y典1/具Y典 具Y典2/具Y典
0.9 1.000 1.000 0.8 0.732 1.268 0.7 0.671 1.329 0.6 0.665 1.335 0.5 0.685 1.315 0.4 0.721 1.279 0.3 0.770 1.230
FIG. 5. Conditional dissipation具⑀Y兩典 from direct simulation of scalar diffusion. Top left:= 0.8. Top right:= 0.7. Bottom left:= 0.5. Bottom right:
= 0.3. Solid line:具⑀Y兩典/具⑀典. Empty circle: 具⑀Y兩典1/具⑀典. Filled circle: 具⑀Y兩典2/具⑀典. Dashed lines: model 具⑀兩典n共具Y 兩典n/兲2/具⑀典.
the mean and variance of the scalar dissipation rate. The latter depends on the turbulence Reynolds number,19
Re1= k
共兲1/2, 共53兲
where k is the turbulent kinetic energy, is the kinematic viscosity, and is the turbulent dissipation rate. If we let environment 2 represent the fluid with below-average scalar dissipation, then we can approximate h2 by a power law of
the form
h2= Re1
−, 共54兲
where 0⬍can be fit to direct numerical simulation(DNS) data.19 For a two-environment model, h1=共1−p2h2兲/p1. The
mean scalar dissipation rate具⑀典 can be modeled using stan-dard methods.1
In Sec. V, we will use direct simulation data for pure diffusion of scalars initialized as a random lamellar system17 to validate a two-environment model with pn= 0.5. As dis-cussed in detail elsewhere,1,17,20because the form of the
sca-FIG. 6. Conditional means from the MECPDF model corresponding to Fig. 2. Top left: = 0.8. Top right: = 0.7. Bottom left: = 0.5. Bottom right:= 0.3.
FIG. 7. Conditional progress variable
具Y 兩典nfor reacting case with Da= 400.
Left: n = 1. Right: n = 2. Line: Ymax.
Circle:= 0.9. Square:= 0.8. Dia-mond: = 0.7. Up-triangle: = 0.5. Left-triangle:= 0.3.
lar PDF and conditional scalar dissipation rates are con-trolled by diffusion, decaying scalar-field statistics for pure diffusion taken at a given value of the mixture-fraction vari-ance closely mimic the corresponding statistics taken from DNS of a decaying scalar field in isotropic turbulence. Thus, using the pure-diffusion data, a preliminary investigation of the validity of the MECPDF modeling assumption can be carried out at a small fraction of the cost of DNS. For the homogeneous two-environment model,具Y 兩典1and具Y 兩典2are
governed by 具Y兩典1 t = SY共具Y兩典1,兲 + 1 2具⑀兩典 2 2共h1具Y兩典 + 具Y兩典1
−具Y兩典2兲 +␥共具Y兩典 − 具Y兩典1兲 共55兲
and 具Y兩典2 t = SY共具Y兩典2,兲 + 1 2具⑀兩典 2 2共h2具Y兩典 + 具Y兩典2
−具Y兩典1兲 +␥共具Y兩典 − 具Y兩典2兲, 共56兲
where h2= 2 − h1. The conditional dissipation rate 具⑀兩典 can
be computed following Girimaji20 by assuming fto be a PDF. Examples of scalar statistics forand⑀can be found elsewhere.17 For pure diffusion in a RLS, the scalar dissipa-tion PDF is nonstadissipa-tionary, and thus Eq.(54) cannot be used to estimate h2. Instead, the values for具⑀典, h1, and␥ will be taken directly from the pure-diffusion data.
For the two-environment model, both reactive and non-reactive cases will be considered. For the non-reactive case, we will use a rate expression of the form5,7
SY共y,兲 = A exp
冋
−共1 − y兲 1 −␣共1 − y兲册
冉
1 −− y 2冊冉
− y 2冊
共57兲 with␣= 0.87 and= 4.0. The preexponential factor A will be set to allow for partial extinction and reignition. For both cases, we first initialize the mixture-fraction field to a double-␦ PDF with 具典 fixed, and then allow it to diffuse until the dimensionless standard deviation =共具⬘
2典/具⬘
2典0兲1/2 equals 0.9. Using this mixture-fraction
field, we then initialize the reaction-progress variable at the reaction-equilibrium value:
具Y兩典n共0兲 = 2 min共,1 −兲. 共58兲
For具典=0.5, this results in an initial mean of 具Y典⬇0.1314, which is conserved for the nonreactive case. Note that
具Y 兩典1共t兲 and 具Y 兩典2共t兲 will evolve differently only if h1⫽1.
For example, if h1⬎1 (i.e., the scalar dissipation rate in
en-vironment 1 is higher than the average), then具Y 兩典1⬍具Y 兩典2
(and vice versa).
V. RESULTS AND DISCUSSION A. Validation of closures
We will first look at the simulation results for pure dif-fusion in a RLS with 具典=0.5 for the nonreactive case. In Fig. 1 conditional statistics and marginal PDFs are shown for several values of . It can be observed that the mixture-fraction PDF is close to aPDF, while the reaction-progress PDF has a much more complicated shape due to the non-trivial initial conditions. The conditional statistics具Y 兩典 and
具⑀兩典 evolve as expected. In particular, 具Y 兩典 remains inside
the upper bound set by the initial conditions(i.e., it is real-izable) and具Y 兩0.5典 approaches 具Y典 for large times (small).
FIG. 8. Conditional progress variable具Y 兩典n for reacting case with Da
= 2000. Left: n = 1. Right: n = 2. Line: Ymax. Circle: = 0.9. Square:
= 0.8. Diamond:= 0.7. Up-triangle:= 0.5. Left-triangle:= 0.3.
FIG. 9. Conditional progress variable具Y 兩典n for reacting case with Da
= 3000. Left: n = 1. Right: n = 2. Line: Ymax. Circle: = 0.9. Square:
= 0.8. Diamond:= 0.7. Up-triangle:= 0.5. Left-triangle:= 0.3.
FIG. 10. Time evolution of the burning index(BI) for selected values of Da. Circle: 400. Square: 2000. Diamond: 3000. The dashed line is the nonreac-tive case with the same initial conditions.
We next look at a priori statistics for a two-environment representation of the pure-diffusion data. In order to define the environments, the pure-diffusion data at each time instant were postprocessed as follows. First, the data were sorted in 50 equal-sized bins in mixture-fraction space. Within each bin, the data were sorted into two equal-number sets (corre-sponding to p1= p2= 0.5) according to the median value of Y.
The first set contained all data less than the median (environ-ment 1), and the second set contained all data greater than the median value (environment 2). These sorted data were then used to compute conditional statistics such as具Y 兩典nfor
each of the 50 bins. Finally, by averaging over mixture-fraction space, unconditional statistics such as 具Y典n were
computed.
Note that other procedures could be employed to define the two environments based on the pure-diffusion data. For example, one need not assume that the probability of each environment is equal to one-half. Likewise, in order to reflect differences in the mixing rate, one could use the median of⑀ within each mixture-fraction bin to sort the data into two equal-sized environments. In order to explore their effects, we have postprocessed the pure-diffusion data using alterna-tive definitions, and have found the results to be consistent with those reported here using the median of Y to define the environments.
Results for具Y典n/具Y典 and hnfor selected values ofare
shown in Table I. Note that, as expected, h1⬎1 and h2⬍1
for⬍0.9. Likewise, 具Y典1/具Y典⬍1 and 具Y典2/具Y典⬎1. These
results confirm the hypothesis that the higher scalar dissipa-tion rate in environment 1 leads to faster mixing, and vice versa. Note that initially具Y典1and具Y典2move away from each
other, as would be expected from the values of h1 and h2. However, near = 0.5, they begin to approach each other. Given the forms of Eqs.(55) and (56), this would only occur when the micromixing term共␥兲 is larger than the diffusion term. Thus, we can conclude from this observation that the micromixing term that describes interactions between the two environments cannot be neglected. Table I also shows two statistics that represent the magnitude of the conditional fluctuations: 1共兲 =
冉
具Y兩典1 2 +具Y兩典2 2 2具Y兩典2 − 1冊
1/2 共59a兲 and 2共兲 =冉
具Y2兩典 具Y兩典2− 1冊
1/2 . 共59b兲The midpoint conditional standard deviation of Y is given by
2共0.5兲, while1共0.5兲 gives the same statistic from the
two-environment representation. The fact that they are not ex-actly equal deserves comment.
When applying QMOM,1,14the conditional means in the environments are chosen(for pn= 0.5) such that
具Y兩典1+具Y兩典2= 2具Y兩典, 共60兲
具Y兩典1 2
+具Y兩典22= 2具Y2兩典, 共61兲 where the conditional moments on the right-hand side are computed directly from the data. In this case,1=2. Instead
of using QMOM, we have defined 具Y 兩典n using the mean values of all data falling above/below the median. Obviously, the two methods are not equivalent. The environment-average(EA) method is seen to underestimate the standard deviation by approximately 26%–39%. The QMOM results can be recovered from the EA values by increasing the dis-tance of具Y 兩典nfrom the conditional mean:
具Y兩典n QMOM =具Y兩典 +2共兲 1共兲 共具Y兩典n EA −具Y兩典兲. 共62兲 Example plots for selected are shown in Fig. 2. Corre-sponding plots for 具⑀兩典n, computed with the EA method,
are shown in Fig. 3. From Fig. 2 we can observe that the differences between the EA and QMOM methods are rela-tively small. We can also observe that initially the higher scalar dissipation rate in environment 1 causes the curves to spread apart. However, at later times Y becomes nearly inde-pendent of and micromixing in Y space causes the two curves to approach each other. Similar behavior is seen in Fig. 3 for the conditional scalar dissipation rates. Also note from Fig. 3 that the three curves have very similar shapes. This justifies the approximation 具⑀兩典n= hn具⑀兩典, where hn is independent of.
We now turn to validation of the models for具⑀Y兩典nand
具⑀Y兩典n given in Eqs. (28) and (26), respectively. Selected
examples are shown in Figs. 4 and 5. Note that in order to reduce the noise in the computation of the derivatives that appear in the models, ten realizations of the scalar diffusion equation were run with different random initial fields, and
具Y 兩典n was found by averaging the ten realizations. In
gen-eral, as seen in Fig. 4, the model for具⑀Y兩典n is reasonably accurate for all values of. In contrast, as seen in Fig. 5, the model for具⑀Y兩典nunderpredicts the data forvalues near 0.5
and overpredicts near the peaks at early times. Indeed, since the derivative is null at the midpoint, the model predicts zero mixing in the Y direction at the midpoint. This assumption becomes poorer at later times where Y is nearly independent of . This mismatch between the model and data was the motivation for adding the conditional IEM model to describe mixing in Y space in Eq.(44). In fact, we can use the value of具⑀Y兩0.5典 to estimate ␥:
␥= 具⑀Y兩0.5典 2具Y兩0.5典222共0.5兲=
CY具⑀典
2具
⬘
2典. 共63兲 The corresponding values for CY are shown in Table I. Not-ing that the values for longer times (smaller ) are more reliable due to the lower correlation between Y and, we can conclude that CY⬇1. This result is not altogether unexpected since both scalars start with the same length-scale distribu-tion and have equal molecular diffusivities.In conclusion we have shown that, although they are not exact in certain details, the models for the conditional joint dissipation rates capture the essential features of the pure-diffusion data. The addition of Gn and Mn can be seen as
correction terms needed to ensure that the closures are con-sistent. Ideally, one could use the pure-diffusion data to vali-date these correction terms. However, except for finding a model for␥as done above, this would not be straightforward exercise. Thus, instead, we will test a posteriori predictions of the two-environment model introduced in Sec. IV against the pure-diffusion data.
B. Nonreactive case
For the nonreactive case, the MECPDF model is given by Eqs.(55) and (56) with SY= 0. For comparison with the pure-diffusion data, we will use Eq.(63) for ␥ and set CY = 1. For homogeneous cases, the conditional scalar dissipa-tion rate can be modeled by17
具⑀兩典 = 具⑀典
1.6共1 −兲1.6
具1.6共1 −兲1.6典, 共64兲
which closely agrees with the expression derived by Girimaji.20 (An alternative model, valid for homogeneous and inhomogeneous flows, has recently been proposed by Devaud, Bilger, and Liu.21) Taking具⑀典/具
⬘
2典 to be constant,the time variable can be rescaled. The only remaining param-eter to fix in the model is h1. From Table I we can observe
that for the pure-diffusion data, h1 is in the range 1.2–1.5.
Using the highest value, we find that the minimum value of
具Y1典/具Y典 predicted by the model is slightly large. We have
thus increased h1 to 1.6 to improve the prediction. Sample
results are shown in Table II and should be compared to the corresponding results in Table I. Overall, the agreement is very satisfactory. Plots of the conditional means are shown in Fig. 6, and can be compared to those in Fig. 2. Again, the agreement between the model and pure-diffusion data is very encouraging. Numerically, the MECPDF model was found to be robust and as easy to solve as the CMC model for this case.
C. Reactive case
For the reactive case, Eq. (57) is used as the chemical source term for Y. All other parameters are taken to be the same as in the nonreactive case. We make the model equa-tions dimensionless by defining= 2具
⬘
2典/具⑀典 and t=t*. A Damköhler number for the reaction can then be defined as Da=A. For the case of forced turbulence with exponential decay of the mixture-fraction variance,(and thus Da) will be constant. As discussed elsewhere,5for small values of Da the reaction will extinguish. In the MECPDF model, extinc-tion will result in both environments dropping well below the reaction-equilibrium curve[Eq. (58)]. At intermediate values of Da, only the environment with the high conditional scalar dissipation rate (corresponding to h1 or environment 1 inTable I) will extinguish at short times. However, at long times, it will be reignited due to micromixing with environ-ment 2. Finally, for large values of Da, both environenviron-ments will remain near the flamelet solution.
Sample results for the reacting case are shown in Figs. 7–9. The curve labeled Ymax corresponds to Da=⬁. In the
first figure, it can be seen that Da= 400 leads to complete
extinction of the reaction. Note that, as discussed above, en-vironment 1 extinguishes first, followed by enen-vironment 2. At later times, the two curves approach each other due to micromixing between environments as seen for the nonreac-tive case. In Fig. 8, the reaction rate is increased to Da = 2000 and partial extinction occurs. For this case, environ-ment 1 drops to the pure-mixing region by= 0.7, but then reignites due to micromixing by= 0.3. Note that environ-ment 2 for this case also drops at= 0.8, but quickly moves back to the flamelet solution at longer times. In Fig. 9 results for Da= 3000 are shown. For this case environment 1 exhib-its a modest drop below the flamelet solution at= 0.8, but quickly recovers. For larger and larger values of Da, this drop in environment 1 will be smaller and smaller.
The degree of extinction and reignition can be quantified using the burning index3,7defined by
BI = 具Y兩0.5典
具Y兩0.5典⬁
, 共65兲
where具Y 兩0.5典⬁= 1 corresponds to the value for Da=⬁. Plots of BI versus t* at selected values of Da are shown in Fig. 10. The dashed line in the figure corresponds to the nonreactive case and establishes the lower bound on BI. As expected from the results shown earlier, the burning index exhibits a clear transition from rapid extinction at Da= 400 to partial extinction and reignition at Da= 2000 and to fully burning at Da= 3000.
The ability of the MECPDF model to predict partial ex-tinction and reignition can be contrasted to the CMC and laminar flamelet models. Because the latter two models only provide a description of the conditional mean progress vari-able, reignition cannot occur in a homogeneous system with stationary turbulence. For the MECPDF model, reignition is made possible by interactions between the extinguished en-vironment and the flamelet enen-vironment. Note also that by adding more environments the transition from the fully burn-ing state to partial extinction and reignition can be captured with increasing accuracy. The MECPDF model offers the additional advantage that a consistent formulation for inho-mogeneous flows is available[Eqs. (49) and (50)] and can be implemented in standard computational fluid dynamics
(CFD) codes for describing partially premixed combustion.
VI. CONCLUSIONS
In this work we have introduced and validated a condi-tional PDF model for describing inhomogeneous turbulent reaching flows based on a consistent extension of the condi-tional moment closure using the direct quadrature method of moments. The derivation begins with the transport equation for the joint PDF of a reaction-progress variable and the mixture fraction. During the course of the derivation, un-closed terms involving the conditional joint scalar dissipation rates appear and are closed by invoking models consistent with those used in the flamelet and CMC models. These closures are validated using data from the direct simulation of the scalar diffusion equation for two correlated scalars. For a two-environment representation, the proposed closures for 具⑀Y兩典nand具⑀兩典n are shown to be in good agreement
with the pure-diffusion data. On the other hand, the agree-ment for具⑀Y兩典n is(as expected) not as good due to incon-sistencies that arise when the system is far from the flamelet regime. In order to correct these inconsistencies, simple cor-rection terms are proposed and validated by comparing MECPDF predictions to pure-diffusion data for 具Y 兩典n.
Fi-nally, the two-environment version of the MECPDF model is employed to describe reactive mixing with different degrees of extinction and reignition as based on the Damköhler num-ber. Overall, the MECPDF model captures qualitatively the dependence of the burning index on the value of the Damköhler number as has been reported from DNS.5,7
In work to be reported in a future communication, we are currently investigating the ability of the MECPDF model to reproduce quantitatively DNS results for reactive scalar mixing.5,7This study seeks to answer such questions as how many environments are needed to capture the essential phys-ics, should the micromixing rate␥depend on the Damköhler number, and how does hn depend on the Reynolds number and the number of environments? In a separate work, we are also investigating the implementation of the inhomogeneous MECPDF model in a CFD code. There, the principal focus is on the relative importance of spatial transport, reaction, and micromixing on the temporal and spatial evolution of extinc-tion and reigniextinc-tion in reacting flows with different local re-action rates. Finally, we note in closing that the MECPDF model can be implemented in large-eddy simulations of tur-bulent reacting flows in a straightforward manner.
ACKNOWLEDGMENT
This work was partially supported by the National Sci-ence Foundation(Grant Nos. CTS-9985678, CTS-0336435).
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